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Theorem lautcnv 40093
Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
Hypothesis
Ref Expression
lautcnv.i 𝐼 = (LAut‘𝐾)
Assertion
Ref Expression
lautcnv ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)

Proof of Theorem lautcnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 lautcnv.i . . . 4 𝐼 = (LAut‘𝐾)
31, 2laut1o 40088 . . 3 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4 f1ocnv 6859 . . 3 (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
53, 4syl 17 . 2 ((𝐾𝑉𝐹𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6 eqid 2736 . . . 4 (le‘𝐾) = (le‘𝐾)
71, 6, 2lautcnvle 40092 . . 3 (((𝐾𝑉𝐹𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
87ralrimivva 3201 . 2 ((𝐾𝑉𝐹𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))
91, 6, 2islaut 40086 . . 3 (𝐾𝑉 → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
109adantr 480 . 2 ((𝐾𝑉𝐹𝐼) → (𝐹𝐼 ↔ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (𝐹𝑥)(le‘𝐾)(𝐹𝑦)))))
115, 8, 10mpbir2and 713 1 ((𝐾𝑉𝐹𝐼) → 𝐹𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142  ccnv 5683  1-1-ontowf1o 6559  cfv 6560  Basecbs 17248  lecple 17305  LAutclaut 39988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-laut 39992
This theorem is referenced by:  ldilcnv  40118
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